For example, I have an alert that was raised stating a database is overdue for a database integrity check. The alert was raised on 1/28 at 11 am. however, on 1/28 10:45pm a database integrity check was done on all the databases within the server and the alert hasn't ended.

when will this alert be fired again in order to change the status from active to ended?

The alert should change state to Ended once that SQL Monitor detects that an integrity check is younger than the threshold set in the alert configuration. This detection will happen the next time that the Base Monitor service runs the DBCC Info (it looks at the result for dbi_dbccLastKnownGood).

A quick test is to run the query described in the alert description and then check the above object (which is also described in the alert).